350 XV NORMED ALGEBRAS AND SPECTRAL THEORY then it is clear thatfxo is a linear form on A, and we have f^s**) = (U(s*)U(s) - x0 | jc0) = (J7(s) • x0 | U(s) • *0) = by virtue of (15.5.1). The corresponding Hilbert form is (15.6.5) gxo(s, t) = (ii(s) • x0 | u(i) - x0). For example, if A = J^(H) where H is a Hilbert space of finite dimension n, and 1A : T\-+T is the identity representation of A in H, then the form/xo is calculated explicitly as follows : if (e^ ^ t^n is an orthonormal basis of H, and n (f f;) the matrix of T with respect to this basis, then for x0 = Y l,ef we have i=l (15.6.6) /JTO-EWv- *»J When studying positive linear forms of the type (15.6.4) we may always assume that x0 is a totalizer for U (15.5), because/^ is unchanged when we replace U by its restriction to the stable subspace of H which is the closure of the stable subspace generated by x0 and the U(s) • x0 (s e A). Under this additional assumption, the form fXQ determines the representation U up to equivalence : (15.6.7) Let U, U' be two representations of A in Hilbert spaces H, H', respectively, and suppose that U (resp. U') has a totalizer x0 (resp. x'0). Then if (U(s) * x0 1 XQ) = (U'(s) - XQ | x'o)for all s e A, the representations U and U' are equivalent. For all s, tin A we have (15.6.7.1) Since the vectors U(t) • x0 (resp. Uf(t)-xf0) form a dense subspace H0 (resp. HO) of H (resp. H'), this proves already that U(s) - x0 = 0 if and only if U'(s) 'x'0 = Q. It follows that, for each z e H0 and each s e A such that U(s)-x0 = z, the vector U'(s)-xr0 is constant and equal to say z' = T- zeHJ). By (15.6.7.1) the mapping Tis an isomorphism of the pre- hilbert space H0 onto the prehilbert space H'0 , which extends uniquely to an isomorphism (also denoted by 7") of the Hilbert space H onto the Hilbert space H' (by virtue of (5.5.4) and the principle of extension of identities). Ite topologi-